On two other occasions, I tried to talk aboutthe fact that a typical recursively-generatedlanguage for a classical bivalent logic couldbe structured as a minimal Hausdorff topology.

Such a topology is feebly compact.

A collection of expressions, organized tosatisfy the description of a topology, cannotbe compact since the language must also supportthe representations of inconsistent sets ofexpressions.

Feeble compactness is strong enough to groundthe compactness theorem with regard tosatisfiable collection. It is weak enoughto admit the use of Koenig's lemma to identifyfinite sets of inconsistent formulas from anarbitrarily given inconsistent set.

Koenig's lemma is needed -- or, possiblyjust the weak Koenig's lemma -- so thatsuch a finite set may be obtained using asystematic semantic tableaux as describedby Smullyan.

====

In "A Formalization of Set TheoryWithout Variables" Tarski and Givantgive five axioms describing the turnstilerelation for an uninterpreted syntacticformalism.

What follows is some thoughts with respectto those axioms.

Tarski and Givant are careful to restrictthe notion of a formalism to the kind ofsystems they are considering. But, theirgeneral description is rather simple:

"..., but also to formalisms *G* providedexclusively with syntactical components, andthus construed as ordered pairs,

< *G*, |- >.

"Such formalisms are often referred to assyntactical or uninterpreted formalisms(as opposed to the original or interpretedformalisms). We can thus arrive at an abstracttheory of syntactical formalisms, i.e., thetheory of all ordered pairs

< *Sigma*, |- >

(where |- is a binary relation between subsetsof *Sigma* and elements of *Sigma*."

I will be interpreting subsets of the systemas global functions on the domain. Those withsignificant knowledge of the lambda calculusmay see similarities of which I am unaware. Iam considering these axioms in other contexts.In particular, I shall interpret these globalfunctions as "selections".

Let X and Y be in *Sigma*. Let Phi, Psi, andDelta be subsets of *Sigma*.

Their first axiom is

"If ( X in Phi ), then ( Phi |- X )"

Topologically, this describes a neighborhoodfilter about X. That is, one has

{ X } |- X

{ X, A } |- X

{ X, A, B } |- X

{ X, A, B, C } |- X

{ X, A, B, C, ... } |- X

with respect to every pairing,

{ X } |- X

{ X, B } |- X

{ X, B, C } |- X

{ X, B, C, ..., A, ... } |- X

and so on.

For the purpose of expressing thisinterpretation of the axiom in thesense of a topological neighborhood,consider rewriting the axiom as,

"If ( {}_f in nbhd(X) ), then ( {}_f accepts X )"

There will be some complexity associatedwith this topological notion which iscomarable to what is found in the settheory axioms. But, it is best to proceedto the additional axioms before discussingthe issue.

In terms of finitary well-formed formulas forwhich the reflexive relation of axiom 1 isexcluded, this corresponds to a directed acyclicgraph.

Now, consider this in the context of Zermelo-Fraenkelset theory for a moment. The axiom of foundationexpresses the fact that the universe of sets ispartitioned into those with the empty set as anelement and those without that member. This isnot a standard characterization of that axiom.But, when the empty set is an element of a set,the axiom is trivially satisfied.

This is easy to see with the initial exampleabove. Consider the difference between thepresentation,

{ X } |- X

{ X, A } |- X

{ X, A, B } |- X

{ X, A, B, C } |- X

{ X, A, B, C, ... } |- X

and the presentation,

{ A } |- X

{ A, B } |- X

{ A, B, C } |- X

{ A, B, C, ... } |- X

With respect to interpreting X as the emptyset, the turnstile relation may be interpretedas the converse of

"X is an element of the transitive closure of Y"

But, while the theory of pure sets has onlythe empty set as a set without members, asyntactic system has an antichain of incomparablesyntactic atomic well-formed parts.

Suppose, now, that one tries to rewrite thesecond axiom as discussed earlier. Then onehas something along the lines of

To get a better sense of what is intended here,consider reading "accepts" as "selects". It iscertainly true that this reading as a "selection"does not quite seem correct since the initialsegment of a deduction could have severalpossible successor steps on the basis oftransformation rules.

But, that is the point. The characterizationgiven by Tarski and Givant is from subsetsto individuals. At this level ofabstraction, those dependencies are obscuredand the specification is that of a fixing asingle element from among alternatives. Thesense of "selection" corresponds withunspecified modal alternatives. The senseof "acceptance" corresponds with a uniqueselection for any given instantiation.

expresses the fact that the transitive closureoperation on a directed acyclic graph yields astrict partial order. I have promoted the ideathat a single formal expression could suffice inthe construction of theories. That statement isschematized by this expression conveying anirreflexive transitive order,

On the other hand, it is closely related tocertain ideas in the foundations of mathematicstoward which I am inclined to have some minorinterest. And, I am personally uninterested infoundational perspectives based uponmetaphysically grounded distinctions.

This is a particular construction with awell-defined complexity and appropriatefeatures for mathematical discourse of allvariety.

The difficulty mentioned earlier begin toappear when we recognize that the first axiomyields expressions of the form,

{ X } |- X

This has a great deal in common with Zermelo'soriginal formulation of set theory. His originalversion relates singletons to denotations anduses the sign of equality to relate denotations.This notion had been changed as the formalismsrecommended by Skolem had been implemented.

For the present purposes, one may formulatethe expression

( {}_X=X accepts X )

Interpret this as one will. It can reflectthe "arrow only" interpretation of categorytheory. It can reflect the ambiguous syntaxof point set topology wherein '{x}' isidentified with 'x' as a syntactic stipulation.It can be a graph-theoretic notion ofself-looping. Pick a favorite belief. Thisexposition is not at the level of an objecttheory.

The sense by which induction is inherentlyimpredicative arises with the transitive closurelogic and its fixed point characteristics. Thereflexive case is an augmentation to the strictpartial order given by the transitive closurelogic.

This can be made precise by considering a single "true"instance of the axiom above. Let '<' be used to expressthe asserted relation and let '>=' be used with its usualsense in relation to '<'. One can consider "accepted","undecided", and "omitted" formulas based on an iteration.

First, the the accepted formulas foreach instance describe a pointedset of language parameters. Second,the accepted types are specifying apartial collection of ordered pairssatisfying the diversity relation( ~x=y ) by means of a single orderedpair. Hence, one can partition thediversity relation in such a way thatthe two components of the partitionseparate into one with semantical contentand one without, where the two componentsare also relational converses. Third, theundecided formulas comprise a boundedchain. However, subsequent axioms couldomit or accept formulas according to anypartial order that does not violate theomitted types.

Augmenting this irreflexivetransitive order with a reflexiverelation introduces a verysignificant problem.

Proper filters are characterizedby the finite intersection property.Neighborhood filters only satisfy thisproperty when they are principalfilters with respect to a singleindividual. An anti-chain oforder-theoretic atoms would haveempty intersection.

Boolean logic provides a means ofaddressing this problem for formalset theories. By analogy, the recursiveconstruction of a formal languagesupports the same method.

In set theory, every set is associatedwith a singleton and the axiom ofpairing can form finite sets. In thetypical recursive construction of acountable language, every finitecollection of well-formed formulas canform pairwise connected terms.

In both cases, the failure ofthe finite intersection propertyfor the collection of neighborhoodfilters is compensated by theexistence of an element thatcorresponds with theintension of finite intersectionor finite conjunction. Thiselement exists as an individual inthe theory.

This is not, however, an ad hoc mechanism.In set theory, the axiom of pairing inconjunction with the axiom of unionimplements a directed set structure. Thisis implicit to theories of arithmetic byvirtue of the difference relation.

It is inherently arithmetical, hencemathematical.

To make this work in set theory, however,one needs the axiom of pairing interpretedin the sense of matroid circuits.

For the formal language, one requiressomething along the same lines.Although Boolean logic provides binaryconnectives, the machinery is notsufficiently weak to convey the senseof mereological weakness one finds inset theory.

The notion of pairing which is neededwould be a single connective -- a completeconnective. Given any two formulas, thereis a formula corresponding to its pair. Thatthe pair is interpreted as a truth-functionalconnective is secondary. However, the reasonfor associating pairing with a completeconnective is so that pairing is semanticallyweak until semantically strengthened throughinterpretation.

As I have explained elsewhere, the systemof basic Boolean functions can berepresented in such a manner that a systemcould use a complete connective in justthis manner. In my formulations, the NORconnective has the semantic significance.Consequently, one could formulate thelanguage recursively using the NANDconnective as a purely syntactic construct.

What I describe as mereological weakness forset theory is expressed by the understandingthat the recursive formulation of the languageis supported by the 4096 axioms in the link,

Unfortunately, when one "knows" whatone means in a truth table, one failsto understand that the recursiveconstruction of a language as a purelysyntactic matter ought to be basedupon axioms just as the transformationrules may be formulated axiomatically.Although this is a simple idea, it makesit difficult for using the syntax effectively.These constructions are intended to groundtypical use, not supplant it.

In recursive construction, there is aunary operation for prefixing a primitivequantification symbol. Such prefixing shouldbe compared with the sequence of nestedsingletons in set theory as far as suchan analogy may allow. What is importantin the present discussion is the role ofpairing.

Before turning to the third axiom, thereis one additional issue to be mentioned.The sense by which the finite intersectionproperty is "implemented by proxy" introducesa problem similar to what one sees inalgorithmically random reals. The systemcontains repetitive information. However,the sense of atomic well-formed parts isthat they be able to take on truth valuationswithout dependency.

Without prejudice, I view this as cause forrejecting systems of logical atomism. It doesnot seem that this problem can be solved bysyntactic stipulations. Perhaps I am wrong.

This clearly resembles a statement ofcompactness. Compare the form of thestatement with

"Every open cover has a finite subcover"

That is an open set notion of compactness,and, it can be associated with consistenttheories (in distinction from closed setmodels which may be associated withparaconsistent logics).

However, this is not quite the fact of thematter.

There are two cases. The first is where( Phi |- X ) is true for every X. In orderto handle this case, one needs at least theweak Koenig's lemma. The discussion ofcompactness in Smullyan using Hintikka setsis what is required. A systematic tableauxapplied to an inconsistent set will eventuallyyield a finite tree. However, it must bepossible for the tree to contain an infinitebranch in order to distinguish between asatisfiable collection and an unsatisfiablecollection.

The second case relies on the fact that theinsight of Frege had been to introduce "TheTrue" and "The False" as objects.

has a diagram of an example of the minimalHausdorff topology which makes thisexplanation simple.

The example begins with the Cartesian productof the positive integers Z^+ and the set,

{ -1, -2, -3, ..., w, ..., 3, 2, 1 }

To this set is added 2 arbitrary symbols. Forthe present purposes, we take these symbolsto correspond with 'T' and 'F'.

In addition, we take the 'w' to correspond withthe 'NAND' connective with which the formallanguage is recursively constructed.

Suppose one is given an enumeration of thewell-formed formulas for a language. Then,for the first collection corresponding tothe set above, take each formula P whichcontains no more than one atomic formula andform the pair,

NAND( P, P )

NAND( NAND( P, P ), NAND( P, P ) )

One can now use the enumeration to formthe first set with the less complex formulasin the postive loci and the negated formulasin the negative loci.

For the next collection, choose only thoseformulas P which contain exactly two atomicformulas. Again form the pairs,

NAND( P, P )

NAND( NAND( P, P ), NAND( P, P ) )

and position them at their respective loci.

In this manner, one can use the complexity onatomic formulas to form the minimal Hausdorfftopology for the language elements in theconstruction.

One now has an equisatisfiable subcollectionof the originally generated formulas. Takingthis as the collection *Sigma* provides alanguage with the appropriate topologicalstructure for implementing the third axiom.

This is very similar to the kind of conditioningKolmogorov implemented to formulate notionssurrounding the law of excluded middle importantto intuitionistic and constructive reasoning.In this case, however, it is being done to supportthe topological structure of a classical,compositional logic.

There are two properties which make thisconstruction of a minimal Hausdorff spacesignificant. Such spaces are Hausdorff-closedand semiregular.

Every Hausdorff space can be densely embedded intoan H-closed space, and, in the class of H-closedspaces, there is a particularly importantH-closed extension called the Katetov extension.With respect to this extension, the originalspace is an open set. The Katetov extensionis defined with respect to the free open ultrafilters,its extended elements comprise a discrete closedset, and the Katetov extensions of semiregulartopologies can always be embedded into a minimalHausdorff topology.

Since every Hausdorff space can be embedded as anowhere dense subspace of a semiregular space, everyHausdorff space can be embedded into a minimal Hausdorfftopology as a nowhere dense space. With respect toforcing models, every semiregular space can be denselyembedded into a minimal Hausdorff topology.

Consequently, a minimal Hausdorff topology canembed densely into itself. This is characterizedsyntactically by the methods discussed above involvingsingletons, pairs, and finite intersections or finitemeets.

To the extent that the third axiom expressescompactness, it is feeble compactness.

A space is feebly compact if every locally finitecollection of open subsets of X is finite. ForH-closed sets, this translates into the properties,

1.

For every open cover of X, there is a finitesubfamily whose union is dense in X

2.

Every open filter has nonvoid adherence

3.

Every open ultrafilter on X converges

Notice that the first statement is not compactness.The compactness theorem in logic is stated onlywith respect to satisfiable formulas whereas thisconstruction involves the structure of uininterpretedsyntax. But, consistent logics are associated withtopological semantics given by open topologies. So,feeble compactness expresses the sense of compactnessin terms of open sets. But, it cannot express thenotion of a finite cover since a consistent theorycannot include every formula.

The second statement reflects the fact that aconsistent theory cannot have an empty domain instandard first-order semantics.

The third statement reflects the notion aboveconcerning fixed point logics.

For minimal Hausdorff topologies, one has,in addition:

Every open filter with a unique adherentpoint converges.

I would presume this corresponds to beingable to assume a first expression as beingtrue. For typical constructions basedupon "ontological" notions of identity, thatmight be the formula,

Ax( x=x )

for example.

It may also correspond with the weak weakKoenig's lemma in the sense that a finitetree is pruned with increasing length. Hence,there is a sense by which infinite length isapproached by asymptotically fewer branches.In particular, the axiom above would havea tableaux consisting of one infinite branch.

However, the stronger notion is required todeal with the inconsistent case. One cannotknow which branches of a systematic semantictableaux can be pruned until they are closed.No asymptotic condition can be given in advance.

All of these topological relations may be foundin "Extensions and Absolutes of Hausdorff Spaces"by Porter and Woods.

To the extent that these topological notionstranslate into consistent theories, I have elsewhereproven that a consistent first-order theory is associatedwith a proximity on its langauge terms. Awodey hasshown that necessary denotations are continuousdenotations in topological semantics for first orderlogic. To the extent that one is speaking of theturnstile relation in the present context, necessarydenotations may be considered to be in correspondencewith eliminable definite descriptions.

"For each Y there exists {}_i such that( {}_i in nbhd(X) ) for finitely many Xand such that ( {}_i accepts Y )"

Taken together, these axioms seem to assert theexistence of an omega-regular filter. A filterF over a domain D is an omega-regular if and onlyif there exists a set E such that ( E subset F ),( |E| = |omega| ), and for each ( d in D ) thereexist only finitely many ( e in E ) such that( d in e ).

Every omega-regular filter is countably incomplete.

Since every open filter has non-void adherence, thiscannot be an open filter.

But, such a filter is necessary. Seemingly, thisfilter would correspond to the filter associatedwith fallacies. By the nature of omega-filters,to the best that I can discern, this filter wouldcontain every finite set.

The existence of this filter has a number ofconsequences as discussed in Chapter 4 of Changand Keisler. I have surveyed the various statementsand found nothing obviously unsupportable with sucha construction interpreted in this way.